In a certain A.P the 24 ^thterm is twice the 10 ^thterm.Prove that the 72 ^ndterm is twice the 34 ^thterm.

To prove: T ₇₂ = 2T ₃₄

We know that,

T _n = a + (n – 1)d

Therefore, T ₂₄ = a + 23d and T ₁₀ = a + 9d
T ₇₂ = a + 71d
T ₃₄ = a + 33d

Now,

∵ T ₂₄ = 2T ₁₀
⇒ a + 23d = 2(a + 9d)
⇒ a + 23d = 2a + 18d

⇒ 2a – a + 18d – 23d = 0
⇒ a – 5d = 0
⇒ a = 5d                                          (1)

Now let us assume, T ₇₂ = 2T ₃₄
⇒ a + 71d = 2(a + 33d)
⇒ a + 71d = 2a + 66d
⇒ a – 5d = 0
⇒ a = 5d                                          (2)
∵ (2) = (1) ∴ our assumption is true
Hence proved.